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Miguel needs to fix a window screen that is 23 feet above the ground. The ladder he uses makes a 75° angle with the ground. What is the shortest possible length of the ladder if the top of it is 23 feet off the ground? Round to the nearest whole number.

Respuesta :

altavistard

Let L be the length of the ladder, and consider the 75 degree angle.  Thinking in terms of trig, the "opposite side" of the triangle formed by the ladder, the ground and the wall is 23 ft.  Your job is to determine the length of the ladder, which is the same as the length of the hypotenuse of the triangle.

The sine function relates these two lenths:

                     opp. side           23 ft

sin 75 deg = ----------------  =  --------------

                       hyp                    hyp

                                             23 ft                     23 ft

Solving for hyp,   hyp = ----------------------  =  ------------ = 23.8 ft                                                                    sin 75 deg             0.956

The minimum ladder length would be 23.8 ft, which rounds up to 24 ft.

Check:  is (23.8 ft)(sin 75) = 23 ft?  Yes.

fichoh

Using trigonometry, the shortest possible length of the ladder given that the wall height is 23 feets and makes an angle of 23° with the ground ls 24 feets

Recall from trigonometry :

  • SOHCAHTOA

  • The length of the ladder is the hypotenus ;

Using :

  • Sinθ = opposite / hypotenus

  • θ = 75°

Sin(75°) = 23 / hypotenus

Cross multiply :

Hypotenus × 0.9659258 = 23

Divide both sides by 0.9659258

Hypotenus = (23 / 0.9659258)

Hypotenus = 23.81 feets

Therefore, the height of the ladder rounded to the nearest feet is 24 feets.

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